Contractible space

A contractible, or contractible space is considered in the mathematical branch of topology. From the viewpoint of homotopy contractible spaces are regarded as trivial. Many invariants defined in algebraic topology disappear for contractible spaces.

Definition

A topological space is called contractible or contractible if it is homotopy equivalent to a one-point space, ie if there is a continuous map

And a fixed point is so

• For all and
• For all

Applies.

Example

The Euclidean space is contractible: Set

Note that the space " is deformed continuously to a point " not in the intuitive sense: The image of Figure

Is the whole space for always, only for the image is only the origin.

Weak contractible spaces

A topological space is called weakly contractible or weakly contractible if for all the homotopy groups are trivial, ie

If a space is contractible, then it is also weakly contractible.

For all the following and in that the CW - complex is contractible: CW - complexes the converse applies. For arbitrary topological spaces the converse is iA do not.

Counter-examples

• The unit sphere ( or more generally a corresponding sphere with fixed radius ) is not contractible, although it is connected for easy.

• The space that you may, as an association of